Neutrosophic Beta Distribution with Properties and Applications

ABSTRACTIn this paper, we discuss stochastic comparisons of the smallest and largest order statistics from independent heterogeneous exponential–Weibull random variables. Let be independent random variables with Further, let be another set of independent random variables with First, when and a matrix with different parameters changes to another matrix in the sense of multivariate chain majorization and row majorization, we investigate the usual stochastic order of the largest order statistics. Next, when and , we establish the usual stochastic order of the largest and smallest order statistics. Finally, we provide sufficient conditions for the hazard rate order of the smallest order statistics.

Moments of order statistics and k-record values arising from the complementary beta distribution with application

For proper actualization of the phenomenon contained in some lifetime data sets, a generalization, extension or modification of classical distributions is required. In this paper, we introduce a new generalization of exponential distribution, called the generalized odd generalized exponential-exponential distribution. The proposed distribution can model lifetime data with different failure rates, including the increasing, decreasing, unimodal, bathtub, and decreasing-increasing-decreasing failure rates. Various properties of the model such as quantile function, moment, mean deviations, Renyi entropy, and order statistics. We provide an approximation for the values of the mean, variance, skewness, kurtosis, and mean deviations using Monte Carlo simulation experiments. Estimating of the distribution parameters is performed using the maximum likelihood method, and Monte Carlo simulation experiments is used to assess the estimation method. The method of maximum likelihood is shown to provide a promising parameter estimates, and hence can be adopted in practice for estimating the parameters of the distribution. An application to real and simulated datasets indicated that the new model is superior to the fits than the other compared distributions

In this paper, a new three-parameter distribution called the Alpha Logarithm Transformed Fr\'{e}chet (ALTF) distribution is introduced which offers a more flexible distribution for modeling lifetime data. Various properties of the proposed distribution, including explicit expressions for the quantiles, moments, incomplete moments, conditional moments, moment generating function R\'{e}nyi and $\delta$-entropies, stochastic ordering, stress-strength reliability and order statistics are derived. The new distribution can have decreasing, reversed J-shaped and upside-down bathtub failure rate functions depending on its parameter values. The maximum likelihood method is used to estimate the distribution parameters. A simulation study is conducted to evaluate the performance of the maximum likelihood estimates. Finally, the proposed extended model is applied on real data sets and the results are given which illustrate the superior performance of the ALTF distribution compared to some other well-known distributions.

The purpose of this study was to compare beta and Weibull distributions in describing basal area diameter distributions in stands dominated by Scots pine and Norway spruce. The material of the study consisted of 535 stands located in eastern Finland. Parameters for both two‐ and three‐parameter approaches of the Weibull distribution were estimated using the method of maximum likelihood. Models for these parameters were derived using regression analysis. For the beta distribution, regression models were formed for the minimum, maximum and standard deviation of diameters within individual stands. These models were used when the exponents of the beta distribution were calculated analytically. Also, some parameter models for beta and Weibull distributions from previous studies were compared with the measured diameter distributions. The distributions obtained were compared using diameter sums and an estimate of the proportion of sawtimber. The results did not reveal any major differences between the suitability of the beta and two‐parameter Weibull distributions. There are appropriate models available for both of the distributions and the more similar the original data is to the data of an application, the better are the results. The two‐parameter approach of the Weibull distribution gave better results than the three‐parameter approach. The poorest results for all the predicted distributions were obtained at the extremes of the distributions.

Comparison of methods for the estimation of statistical parameters of censored data

<p>Order statistics is a significant research topic within probability and statistics, particularly due to its widespread application in areas such as reliability and actuarial science. Extensive research has been conducted on extreme order statistics, and this paper focused on the second-order statistics. Specifically, the study investigated the second-largest order statistics derived from dependent heterogeneous modified proportional reversed hazard rate samples, utilizing the stochastic properties of the Archimedean copula. This paper first examined the usual stochastic order of the second-largest order statistic between two groups of dependent heterogeneous random variables. These variables were analyzed under conditions involving the same tilt parameters with different proportional reversed hazard rate parameters, and different tilt parameters with the same proportional reversed hazard rate parameters. The study derived the sufficient conditions required for establishing the usual stochastic order in these cases. Next, the paper addressed the reversed hazard rate order relationship for the second- largest order statistic between two groups of independent heterogeneous random variables. This analysis was conducted under various conditions: the same tilt parameters with different proportional reversed hazard rate parameters, different tilt parameters with the same proportional reversed hazard rate parameters, and different sample sizes with the same parameters. The sufficient conditions for establishing the reversed hazard rate order were also derived. Finally, the theoretical findings were substantiated through numerical examples, confirming the main conclusions of the paper.</p>

The aim of this paper is to highlight the use of Zero inflated Poisson distribution and Hurdle Poisson distribution to improve goodness of fit of data with excess zeros. Claim frequency data were used for the Singapore motor insurance database available on the internet. The data are tested for the detection of excess zeros. Parameters of Zero inflated Poisson distribution are estimated using method of moments and method of maximum likelihood. Mean and variance formulas are derived for the distribution of Hurdle Poisson and estimators of method of moment and maximum likelihood of unknown parameters are derived. Parameters of Hurdle Poisson are estimated. The data are modeled using Poisson distribution, Zero inflated Poisson distribution and Hurdle Poisson distribution. Goodness of fit are tested using Chi Square test. The best distribution in the study is selected using AIC & BIC criteria. Hurdle Poisson is the best distribution for modeling the data.

We first prove a uniquenes property of the Uniform distribution on the unit interval and then establish its charaterization based on the structure of the expected spacings between the consecutive order statistics as also through the independence between the largest order statistics as also through the independence between the largest order statistic X(n) and the ratio [Xbar]/X(n), [Xbar] being the sample mean.

Ordinal data are often modeled using a continuous latent response distribution, which is partially observed through windows of adjacent intervals defined by cutpoints. In this paper we propose the beta distribution as a model for the latent response. The beta distribution has several advantages over the other common distributions used, e.g. , normal and logistic. In particular, it enables separate modeling of location and dispersion effects which is essential in the Taguchi method of robust design. First, we study the problem of estimating the location and dispersion parameters of a single beta distribution (representing a single treatment) from ordinal data assuming known equispaced cutpoints. Two methods of estimation are compared: the maximum likelihood method and the method of moments. Two methods of treating the data are considered: in raw discrete form and in smoothed continuousized form. A large scale simulation study is carried out to compare the different methods. The mean square errors of the estimates are obtained under a variety of parameter configurations. Comparisons are made based on the ratios of the mean square errors (called the relative efficiencies). No method is universally the best, but the maximum likelihood method using continuousized data is found to perform generally well, especially for estimating the dispersion parameter. This method is also computationally much faster than the other methods and does not experience convergence difficulties in case of sparse or empty cells. Next, the problem of estimating unknown cutpoints is addressed. Here the multiple treatments setup is considered since in an actual application, cutpoints are common to all treatments, and must be estimated from all the data. A two-step iterative algorithm is proposed for estimating the location and dispersion parameters of the treatments, and the cutpoints. The proposed beta model and McCullagh's (1980) proportional odds model are compared by fitting them to two real data sets.

The design wind pressures specified in the 2005 National Building Code of Canada (NBCC) have been derived from the Gumbel distribution fitted to annual maximum wind speed data collected up to early 1990s. The statistical estimates of the annual maxima method are affected by a relatively large sampling variability, since the method considers a fairly small subset of available wind speed records. Advanced statistical methods have emerged in recent years with the purpose of reducing both sampling and model uncertainties associated with extreme quantile estimates. The two most notable methods are the peaks-over-threshold (POT) and annually r largest order statistics (r-LOS), which extend the data set by including additional maxima observed in wind speed time series. The objective of the paper is to explore the use of advanced extreme value theory for updating the design wind speed estimates specified in the Canadian building design code. The paper re-examines the NBCC method for design wind speed estimation and presents the analysis of the latest Canadian wind speed data by POT, r-LOS, and annual maxima methods. The paper concludes that r-LOS method is an effective alternative for the estimation of extreme design wind speed.Key words: wind speed, extreme value theory, order statistics, return period, maximum likelihood method, peaks-over-threshold method, generalized extreme value distribution, Gumbel distribution, generalized Pareto distribution.

This paper deals with order statistics from a gamma or χ2 (Pearson Type III) distribution. Expressions are derived for the moments of an order statistic and for the covariance between two order statistics. A table of moments (about the origin as well as the mean) is presented. Equations are derived which are used to solve for the percentage points of the order statistics and a table of the percentage points is given. The problem of the best linear unbiased estimate of the scale parameter of the distribution which is based on order statistics is, briefly, discussed for the censored (type II) case. The modal value of an order statistic is derived and a table of these values is given for the cases of the smallest and the largest order statistics. Applications ‘to life-tests, extreme values, reliability and maintenance are described and illustrated in some cases.

In this work, we present a five-parameter life time distribution called Harris power Lomax (HPL)  distribution which is obtained by convoluting the Harris-G distribution and the Power Lomax distribution. When compared to the existing distributions, the new distribution exhibits a very flexible probability functions; which may be increasing, decreasing, J, and reversed J shapes been observed for the probability density and hazard rate functions. The structural properties of the new distribution are studied in detail which includes: moments, incomplete moment, Renyl entropy, order statistics, Bonferroni curve, and Lorenz curve etc. The HPL  distribution parameters are estimated by using the method of maximum likelihood. Monte Carlo simulation was carried out to investigate the performance of MLEs. Aircraft wind shield data and Glass fibre data applications demonstrate the applicability of the proposed model.

The problem of efficient estimation of the parameters of the extreme value distribution has not been addressed in the literature. Our paper attempts to obtain efficient estimators of the parameters without solving the likelihood equations. The paper provides for the first time simple expressions for the elements of the information matrix for type II censoring. We consider the problem of estimation of θ and σ in the model (1/σ)f((x − θ)/σ) when f is the standard form of the pdf of type I (maximum) extreme value distribution. We construct efficient estimators of θ and σ using linear combinations of order statistics of a random sample drawn from the population. We derive explicit formulas for the information matrix for this problem for type II censoring and construct efficient estimators of the parameters using linear combinations of available order statistics with additional weights to the smallest and the largest order statistics. We consider numerical examples to illustrate the applications of the estimators. We also consider a Monte Carlo simulation study to examine the performance of the estimators for small samples. Using the asymptotic covariance matrix of the estimators in the above model we then construct an efficient estimator of θ in the reduced model (1/cθ)f((x − θ)/cθ) where θ ( > 0) is unknown and c ( > 0) is known for complete and censored samples.

Modeling extreme stochastic phenomena associated with catastrophic temperatures, heat waves, earthquakes and destructive floods is an aspect of proactive mitigation of risk. Hydrologists, reliability engineers, meteorologist and researchers among other stakeholders are faced with the challenges of providing adequate model for fitting real life datasets from the extreme natural hazardous occurrences in our environment. Convoluted distributions (CD) and generalized extreme value (GEV) distribution for r- largest order statistics (r-LOS) have been some of the prominent existing techniques for modeling the extreme events. This study explored the properties of order statistics from the convoluted distribution as alternative procedure for analyzing the extreme maximum with the aim of obtaining superior modeling fit compared to some other existing techniques. The new procedure called MAXOS-G employed the potential properties of the Maximum Order Statistics (MAXOS) and the flexibilities of convoluted distributions where G is taken to beWeibull-Exponential Pareto (WEP) and the New Kumaraswamy-Weibull (NKwei) distributions. The maximum order statistics of the WEP distribution (MAXOS-WEP) and NKwei distribution (MAXOS-NKwei) was derived and applied to three datasets consisting of annual maximum flood discharges, annual maximum precipitation and annual maximum one-day rainfall. Some properties of the MAXOS-WEP was investigated including the moment, mean, variance, skewness, and kurtosis. Characterization of WEP distribution by the L-moment of maximum order statistics was presented and coefficient of L-variation, L-skewness and L-kurtosis were derived. The results from the application to three datasets using R-software justified the importance of this new procedure for modeling the maximum events. The MAXOS-NKwei and MAXOS-WEP models provide superior goodness-of-fit to the datasets than the WEP and NKwei distributions and better than some previously proposed convoluted distributions for modeling the datasets.